Boundedness and compactness of commutators associated with Lipschitz functions

2018/01/17 by Weichao Guo, Jianxun He, Guo, Weichao +5 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Banach Space Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1801.06064

Abstract

Let α∈ (0, 1], β∈ [0, n) and T_Ω,β be a singular or fractional integral operator with homogeneous kernel Ω. In this article, a CMO type space \rm CMO_α(\mathbb Rⁿ) is introduced and studied. In particular, the relationship between \rm CMO_α(\mathbb Rⁿ) and the Lipchitz space Lip_α(\mathbb Rⁿ) is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator (T_Ω,β)^m_b on weighted Lebesgue spaces via functions in Lip_α(\mathbb Rⁿ), and an equivalent characterization of the compactness for (T_Ω,β)^m_b via functions in \rm CMO_α(\mathbb Rⁿ) are obtained. Some results are new even in the unweighted setting for the first order commutators.

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